Electronic transport driven by collective light-matter coupled states in a quantum device

In the majority of optoelectronic devices, emission and absorption of light are considered as perturbative phenomena. Recently, a regime of highly non-perturbative interaction, ultra-strong light-matter coupling, has attracted considerable attention, as it has led to changes in the fundamental properties of materials such as electrical conductivity, rate of chemical reactions, topological order, and non-linear susceptibility. Here, we explore a quantum infrared detector operating in the ultra-strong light-matter coupling regime driven by collective electronic excitations, where the renormalized polariton states are strongly detuned from the bare electronic transitions. Our experiments are corroborated by microscopic quantum theory that solves the problem of calculating the fermionic transport in the presence of strong collective electronic effects. These findings open a new way of conceiving optoelectronic devices based on the coherent interaction between electrons and photons allowing, for example, the optimization of quantum cascade detectors operating in the regime of strongly non-perturbative coupling with light.

. Diagram describing the quantum model for infrared photodetector in the ultra-strong lightmatter coupling regime. A quantum cascade detector is coupled with a single mode of a patch microcavity through the quantum Hamiltonian (1)(2)(3). Our aim is to determine the photocurrent Iph induced by an external photon drive Sin. The two ellipses represent the quantum coherences ρ21 and ρ31 which are particularly important for our quantum theory. The scheme of the photodetector highlights the grating coupling the light into the active region.
is an operator that describes the collective electronic polarization between the subbands 1 and 2. The third term of the right hand side of (1) describes dipole-dipole interactions that lead to plasmonic effects and a blue-shift of the 1→2 transition 1 , with ℏ4 8 the plasma energy defined in such a way that: (4) 4 8 ² 6 * :: ; <= >?
This quantity can be regarded as the plasma frequency of a single electron (see also 2), where S is the area of the heterostructure and LQW an effective thickness of the quantum well 1 . Thus, the fourth term describes interaction between the 1→2 transition and the cavity with a single electron coupling constant Ω @ 4 8 /2 √B 1 . Here F=NpLQW/Lcav=0.13 expresses the geometrical overlap between the electron gas and the cavity mode, with Np the number of quantum wells and Lcav the total thickness the patch cavity. The last term of (1) corresponds to the coherent tunnelling of electrons between the levels 2 and 3. In the above Hamiltonian, for simplicity, we neglect the plasmon effects other than the ones arising between transition 1→2. This is justified by the small overlap of wavefuctions i>2 with the fundamental level 1, by design. Also, the photon wavevector of the cavity photon ~$ 23 /ℏ% being typically very small in comparison with the electronic wavevectors k we consider only vertical excitations, which justifies the form for .
A natural framework to express the electronic relaxation is the density matrix approach 3 , where the Hamiltonian evolution is supplemented with relaxation terms. The system is considered at a temperature T=0 K, with no electron above the Fermi level in the fundamental state. To implement this approach, we define electronic populations Ni=∑ 〈% 5 * % 5 〉 5 and coherences ij=∑ 〈% 5 * % E5 〉 5 . The relaxation paths and times for the electronic populations are indicated by black arrows in Fig S1, with the notation ij that is the overall relaxation time from subband i to subband j. Relaxation rates for coherences ij are noted ji; the density matrix equations can thus be written: For the photonic degrees of freedom, we apply a similar approach that is reminiscent of input-output relations. To this end, we introduce coupling between the cavity and an external incoming Sin and reflected field Sout and we introduce coefficients that describe the radiation r and non-radiation loss in the system nr. To express the equation of motion we project the cavity fields into a coherent state )|T > T|T >, and we introduce the two quadratures of the field V T T * and $ T * + T . It turns out that for the complete set of equation it is also convenient to introduce the quadratures for the coherences of the density matrix: Which is a real quantity since M E M E * . We further introduce the following useful quantities: Here N0 is the total number of electrons in the structure. We note that at equilibrium and at T=0 K the Fermi level EF lies just below the second subband.
Assuming zero bias applied on the structure, the full set of evolution equations is: (9)  In the above equations we have assumed parabolic band approximation 4 E 4 . + 4 E. 4 + 4 E , and we have set 4 $ 23 /ℏ. We also assumed a periodic system with N0 electrons per period, i.e. where the level 5 injects electrons to the level 1' of the next period (see Figure S1). Equations (9)-(21) derive from Eq. (5), assuming the absence of correlations between the photon field and the quantum well coherences: i.e. 〈 ) * + ) % 5 * % 5 〉 T * + T 〈% 5 * % 5 〉. We call this condition "semiclassical condition"; it should be revised in the deep ultra-strong coupling where one seeks to establish the link between vacuum field fluctuations and dark current of the detector 4 . Eq. (22)-(24) can be seen as a generalization of the usual coupled-mode theory beyond the rotating wave approximation; they are equivalent to the Maxwell's equation to the cavity field coupled with the electronic polarization, with the requirement to satisfy the energy conservation.
The cavity is driven by an incident field that can be written in complex notations as W g Re W j k . The equations set (7-24) is non-linear and its general solution presents a complex task. However, using simple assumptions we can provide an analytical solution that provides the constant photocurrent Iph induced by the incident photon flux |W j |². The DC photocurrent generation is seen as a rectification effect on the incident harmonic drive 5 W j k . To obtain the DC response of the system, we assume a first order development of the non-linear set (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24), where variables can be developed in a Fourier series: (25) l g l m Re l n k l n k ⋯ Here l m denotes the time average for l g . We consider the lowest order solution where we keep the first non-zero term in the Fourier series (25) for each variable in the equation set (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24). This approach is valid if the detector is probed by a weak source and the absorbing transition is far from saturation.
Let us now identify which is the lowest order for the expansion in (25) for each variable of the problem. Clearly, the field variables E and A have zero means values, therefore the lowest order is $ n and V j . Similarly, eq. (22) indicates that this the case for the coherences W and X which correspond to the oscillating dipole that interacts with the cavity. Next, the populations O have relaxed to their average values O p in a steady state; it can be shown that the next relevant component is O , which leads to nonlinear optical effects (this will be discussed in another paper). This assumption is consistent with the form of equations (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24). For instance, looking at eq. (9) we see that the evolution of O is coupled to the product of two oscillating terms Y · X that will yield a non-zero average. Finally, examining the remaining set of equations we can show that the extractor coherences W / and X / have zero mean value, while the tunneling coherences W / and X / have necessarily non-zero average values W ̅ / and X p / ; those average values allow expressing the 2→3 DC tunneling current in the Kazarinov and Suris approach 6 . In the following, we simplify notations and write the average quantities by l instead of l m and the complex amplitude of oscillating terms l n by instead of l n . We also introduce the following scalar product of two oscillating variables: (26) l g l g mmmmmmmmmmmmmmm st u n .u n * 〈l n · l n 〉 Next, we set all time derivatives of average terms to zero, and all-time derivatives of oscillating terms to 4l n . We arrive at the following set of algebraic equations, where we have eliminated the populations N4 and N5: Here Γ Γ Γ . In the above equations, we have introduced the collective plasma frequency 4 8 and the collective Rabi frequency Ω @ according to the equations: , Ω @ #O [ Ω @ We have supposed that most of the electronic population is on the first subband, O [ ≫ O y , We have neglected the terms YX / and YW / ; this is justified since these terms are negligible as long as saturation effects are not important. The equation set can then be solved analytically, where all quantities can be expressed as a function of the amplitude of the incident field W j .
In a steady state, the electromagnetic energy of the cavity does not vary with time, ]-| n| z|d n | 0 ] 0. We can thus express the energy conservation of the system, and define the frequency dependent reflectivity ! 4 and the absorption efficiency | 4 for the 1→2 transition: It can be shown that these expressions reduce to the usual coupled-mode theory 7 by performing the rotating wave approximation; however, these expressions are more general and can be applied for an arbitrary light-matter coupling strength, and in the presence of strong collective electronic effects 8 . Finally, the photocurrent is defined as the out-going current Y •€ kO / /R / that appears for an incident drive. With this definition, our equations provide a photocurrent that is proportional to the square amplitude of the incident drive, Y •€ ∝ ‚W j ‚ . By setting ‚W j ‚ to be the number of photons per second with an energy ℏ4 we arrive at the following expression for the frequency dependent responsivity of the detector: Here the factor GT depends only on scattering times and the tunneling time t, but not on the energy of the incident photon: b, as stated in the main text. Since 0<GT<1 we can relate GT to the extraction probability pe previously introduced for QCDs 9, 10 . The factor 1/Np is commented further.
The absorption quantum efficiency from Eq.(38) depends on the mean value | 4 ∝ 〈W j . V j 〉 〈 M M . V j 〉. Clearly it depends only on the coherence M of the 1→2 transition and its complex conjugate M . Since V j is proportional to the intracavity field, we can state, as in the main text, that in essence | 4 is proportional to the mean value of product of M and the intracavity field. The explicit expression of the absorption efficiency | 4 is: Here Finally, a very important quantity is the denominator Π 4 in Eq. (41) which is expressed in the following way: The complex zeroes of this denominator provide the frequencies and damping rates of the light-matter coupled states, while the residues of the function 1/Π 4 is connected to the Hopfield coefficients. This will be discussed in details in a subsequent work. Let us consider a case without tunnel coupling Ω 0 and no damping, Γ L 21 0. Then Eq. (47) reduces to Π 4 4 oe + 4 2 4 % 2 + 4 2 + B4 8  Finally, let us provide the expression of the coherent gain function … ‡ 4 : Here we have: The function … ‡ 4 has been discussed in the main text, and its simplified version and physical meaning is discussed in section 1.3 of the supplementary file. These expressions conclude the analytical solution of the problem for a microcavity coupled quantum detector with tunnel extraction.
Let us now examine the limit of the theory in the absence of extractor coherences W j /, X w / . I that case the matrix (45) becomes an identity and we have | 1 and | ] 0. However, we still keep the coherences X / and W / in order to have tunnel extraction from level 2 to level 3. We call this approach "conventional" detector theory. Alternatively, we can set very strong damping L / → ∞ for the coherences W j /, X w / which will produce the same effect. In that case, we find the responsivity is provided by the formula: The expression in the square brackets is essentially equal to 1 for an optimal detector in the weak coupling regime with 4 ≈ 4 ≈ 4 ; we thus recover the well-known result from the literature 9,10,11 .
Let us now comment on the factor 1/Np that appears in Eq.(39) and Eq.(50). In our model, the photocurrent is expressed as Y •€ kO / /R / , where N3 is the population on level 3 on a single period of the structure. In a permanent regime, the current is continuous across the structure, that is an electron exciting on period to the right (5→1', or 5→right contact in Figure S1) is replaced by an electron entering the period from the left (5''→1, or left contact→1 in Figure S1). This is the "Eulerian" picture of the transport in QCD as described by A. Delga in Ref. [10] (page 347). The current Y •€ is therefore the total current circulating in the external read-out circuit when the detector is irradiated with a photon flux |W j |².
In our system of equations we are computing the total absorption of the system, , and the populations are proportional to Thus, for a multiperiod structure, and assuming identical populations Ni of the i th level in all periods, our model actually provides NpN3 instead of the population N3 on a single period. Thus we have to correct with a factor 1/Np in order to obtain the correct value of the current Iph.
Another way to justify the factor 1/Np is the expression of the absorption itself from Eq. Our approach can be further generalized to more complex structures where several occupied subbands are present. Furthermore, it can be extended to study optical-nonlinearities and saturation in the strong coupling regime. These aspects will be presented in future works.
. ℏΩ g ∑ -% 2. † % 3. % 3. † % 2. 0 . . This single particle Hamiltonian can be cast in a matrix form: We can diagonalize this Hamiltonian by diagonalizing the 2x2 matrix defined above: The matrix element of the transformation matrix has been normalized so that the fermionic commutation rules hold for the new operators.
We have determined the tunnel coupling energy ℏΩ for our structure by simulating the main quantum well and the extraction region for various values of the tunnel barrier Lbar, as indicated in Figure S2(a).
In Figure S2(b) we plot the energy differences Ei1 = ħik -ħ1k as a function of Lbar. In the 2x2 matrix approach described above we can infer the coupling constant from the following equation: Which follows directly from Eq.(54), as we expect that for very large values for Lqw we recover the uncoupled energies. The result has been plotted in Figure S2(c) in black dashed curve; this approach yields a value ħt = ħ2,3 = 5.1 meV for Lbar =5nm.
From Figure S2 (b) we see that the Ei1 is affected as well from the tunnel coupling; in order to estimate that effect we have analyzed the energies with a 3x3 matrix of the form: The corresponding coupling constants are plotted as continuous lines in Figure S2 Here the probability current density is computed at the barrier. The results from this expression are plotted as dotted curves in Figure S2(c) and they corroborate the values from the previous estimations: ħ2,3 = 5.1 meV, ħ2,4 = 1.5 meV.
The equation of motion described in 1.1. can also be written in the extended basis described above. In the resulting equation are equivalent to the description in 1.1., but appear in a slightly more complicated form. More general set of equations, which feature an optical transition between subbands 1 and 3 and plasmon-plasmon couplings between 1→2 and 1→3 will be discussed elsewhere.

Discussion of É Ê Ë and a simplified expression
In our model, the photocurrent is expressed from the population of the extractor level N3. It is therefore interesting to examine this variable more closely. Using the equation system (27-36) we have the following expression: The first contribution, 〈Y j · X w 〉, is expressed solely from the coherences M and the photon field. The quantity 〈Y j · X w 〉 is closely linked to the absorption efficiency | 4 ; we can show that they are actually proportional to each other, with a factor that depends weakly on the frequency 4. The second contribution is expressed from the "optical rectification" of the coherence M / and the optical field.
Assuming that X w / ∼ W j / and 4 / ≫ L / we can rewrite (59) as From this form, we can state that there are essentially two pathways to create a population on level 3 by photon absorption. Th first one is an absorption of a photon between 1→2 and then a transfer from level 2 to level 3; this process is weighted by the tunnel gain factor … . This is the process that has been considered so far in the conventional detector theory. Eq.(59) shows a second mechanism, that is linked to the coherence M / (the transition 1→3). In the localized basis chosen for our discussion this transition is not directly excited by light. We can however link the coherence M / to the optically active coherence M by using the equations of motion (20) and (21). Eliminating W j / in favor of X w / the following equation is obtained: The quantity thus satisfies a driven oscillator equation, where the driving term is proportional to the tunnel coupling Ω as well as the coherences M (M ) and their time derivatives. Eventually, we can simplify the source term once again using the approximation X w / ∼ W j / and 4 / ≫ L / , as shown in the last term of Eq.(61). Solving (61) in a rotating wave approximation, we obtain the equality: It is clear form this expression that second term in Eq.(60) is resonantly enhanced at 4 ≈ 4 / and it is proportional to the optical absorption. Eventually, the population N3 is enhanced by the product ¶ z ¨Î¨_ ² Ï _ ² ; this product is maximized when the extractor transition 4 / is matched with the resonance of the optical absorption | 4 , which peaks either at the energy of the collective state in the mesa configuration, or the polariton state in the cavity-coupled detector. Inspired from the above discussion, we provide now a simplified expression for the coherent gain function … ‡ 4 , specifically for the case of large detuning of the extractor 31>>21. A finer approximation is obtained by working directly on the full result from Eq. In the latter expression we have considered the fact that the real part of Δ 4 dominates the imaginary part for frequencies larger than the resonance 21. Next, expanding the numerator of Eq.(49) and simplifying the expression for ¯/ ≫ 1, at the following simplified expression: (64) Here we have added unity to the simplified expression of … ‡ 4 in order to take into account the first term in Eqs.(59,60). In this simplified version, the function … ‡ 4 appears as a Lorentzian function centered at the extractor transition 4 / with an amplitude that grows as the product of the three quality factors (4 /L )(4 / /L / )( 4 / /L / . In Figure S3 this expression is compared numerically with the full expression of … ‡ 4 for typical values of our detector model. The Lorentzian model reproduces qualitatively the spectral dependence of … ‡ 4 very well; the background value, far from the resonance, is also well accounted from the unity in Eq.(63). In particular the behavior of the maximal value of … ‡ 4 at the extractor energy is well reproduced, especially for the case of strong detuning. This model confirms that the strong increase of the maximum … ‡ 4 4 / arise from the increasing product (4 / /L / )( 4 / /L / . The product of the three frequencies 4 4 / 4 / is reminiscent of phenomena encountered in non-linear optics 13 . In the present case it can be argued that the photogeneration of the DC current through light absorption is essentially a rectification phenomenon, where the AC electric field of the incident electromagnetic radiation into a DC current 14 . The non-linearity in this case is provided by the intrinsically non-linear character of the Maxwell-Bloch equations in the fermionic system (Eq.(7-24)), and cannot be recovered from bosonized models. This rectification mechanism, which according to our results is optimized at the extractor frequency 31 allows electron transport at the frequency of the polariton states when UP is resonant with 4 / .

Supplementary Note 2: Fitting parameters for the quantum model
In the quantum model we used to fit the photocurrent we had to provide the value for radiative and nonradiative losses of the cavity. Since we do not have access to a sample with exactly the same layers and zero doping, we decided to simulate such a device. In particular we simulated the reflectance of a ribbon as a function of its width, i.e. changing the cavity resonance, over a sample characterized by a 4 8 0.
In Figure S4 we report on the left the result of such simulation. From the contrast and width of the resonance we can extract the value of the losses as a function of the cavity frequency:

Supplementary Note 3: Growth sheet of the device
Devices were obtained by MBE growth. Complete sequence of the grown layers is provided in table S1. Table S1. Growth sheet of the device investigated in this work. The structure was grown over a GaAs substrate.

Supplementary Note 4:
Other characterization of cavity-coupled detectors: Reflectance, Photocurrent temperature and voltage dependence and Responsivity.

Reflectance
In figure S5a we reported a SEM picture of the photodetectors we fabricated. Each device in the column has different ribbon width w, indicated in microns, and each column is replicated 9 times. The area of photodetection was roughly 100 × 100 μm 2 . The reflectance of each structure is reported in fig S5b. Each spectrum corresponds to the reflection of the TM mode normalized on the TE mode from a grating with roughly 1 mm 2 size. The spectra are stacked on top of each other, from the lowest cavity energy (Þ 2 μm) to the highest (Þ 1 μm). The continuous and dotted lines serve as a guide for the eyes representing, respectively, the polaritonic dispersion, the second order cavity mode and the energy of the extraction channel.

Photocurrent Temperature dependence
Each device was also characterized in temperature. In Figure S6 (left) we report, in example, the spectrum obtained with a photodetector with ridges width of 1.1 μm (~150 meV) at different temperatures. The photocurrent spectrum can be clearly measured up to 60K with a standard FTIR spectrometer. On the right we report the integral of the photocurrent as a function of the temperature, for different cavities (see also Fig.2(e) of the main text). One can see that the highest photocurrent is measured with ridges width of 1.3 μm. That device is indeed the one with the UP resonant with the extraction channel. The photocurrent signal drops exponentially with the temperature, as the electrons escapes the main QW due to thermal effects.

Photocurrent Voltage dependence
By applying a bias to the detector, we observe a bending of the structure inset in Fig. S7. The energy of the extraction channel changes and so does the spectrum of photocurrent. One can see how the intensity of the photocurrent changes to be maximized by the UP branch at 0 bias to the LP at negative bias. This behavior can be reproduced with our model by changing the energy of the 1→3 transition. In particular by keeping the energy E13 as a fitting parameter we managed to reproduce the experimental data with good agreement (right panels of Fig. S7). When applying a positive bias to the structure we observed a more complex behavior: a change of sign in the photocurrent spectra was observed. This suggest the presence of two counter-streaming flows of electrons, one possibly following the extraction channel and the other tunneling above the barrier. Such a phenomenon is not included in our model and for that we limited our comparison to the negative values.

Responsivity
In order to evaluate the responsivity of our detector at 10K we estimated the power emitted by the glowbar of the FTIR, which is the one of a black body of 1200K, that impinges on the sample. The photocurrent value can be measured with a lock-in, and in particular the value will correspond to the integral of the photocurrent spectra; one can see, e.g. Fig.S6, that all the contribution comes from photons with energy comprised between ~120 and ~175 meV. By dividing the spectral radiance of the glowbar in that range and the integral of the photocurrent we estimated a responsivity of ~50 mA/W. The responsivity of our detectors was also tested with a powerful commercial QCL (MIRcat). The emission of the laser was characterized with a detector as a function of the emitted wavelength, Figure  S8 (blue line). Then we shined our device and measured the photocurrent (orange line). The responsivity was calculated assuming that all the power was impinging on our detector, leading to a lower limit of the peak responsivity of ~2 mA/W at 80K. This correspond to ~70 mA/W at 10K, in agreement with the black body measurements.